Qwestrum Engineering360 · Civil Engineering · Soil Mechanics
Earth Pressure Theories
Pick the coefficient for the wall movement — active K_a = tan²(45 − φ/2) if the wall moves away, passive K_p = tan²(45 + φ/2) if it pushes into the soil — then integrate the pressure over the height to get the thrust P = ½KγH².
Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.
Key formulas & points
Skim these first — then read the full notes below.
- Active: wall moves away; passive: wall moves toward soil
- Cohesion: modify pressure diagram — tension crack possible at top
- At-rest K_0 ≈ 1 − sin φ for normally consolidated soil
Topic details
Introduction
Earth pressure theory predicts the lateral thrust that soil exerts on retaining structures, and the value depends critically on how the wall moves. A wall yielding away from the backfill mobilises the minimum active pressure; a wall pushed into the soil mobilises the maximum passive resistance.
Scope in B.Tech and GATE syllabus
Rankine’s theory assumes a smooth vertical wall and gives simple coefficients K_a and K_p from the friction angle. Coulomb’s theory is more general, allowing wall friction, wall inclination and sloping backfill, and uses a trial-wedge or analytical approach.
Why this topic matters in practice
For cohesive backfills the active pressure diagram includes a negative (tension) zone near the top, leading to a tension crack; this modifies the thrust and its point of application, a detail examiners frequently test.
Key relations & formulas
Formulas (Indian textbook notation)
Formulas (Indian textbook notation)
(horizontal, level backfill, no surcharge)
Formulas (Indian textbook notation)
Notation and sign conventions
Relation 1 —
Formulas (Indian textbook notation)
Write this relation with symbols exactly as in Soil Mechanics & Foundations — BC Punmia before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
Formulas (Indian textbook notation)
Write this relation with symbols exactly as in Soil Mechanics & Foundations — BC Punmia before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
(horizontal, level backfill, no surcharge)
Write this relation with symbols exactly as in Soil Mechanics & Foundations — BC Punmia before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 4 —
Formulas (Indian textbook notation)
Write this relation with symbols exactly as in Soil Mechanics & Foundations — BC Punmia before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Fundamentals and definitions
The at-rest state K_0 ≈ 1 − sin φ describes soil that has not moved, relevant to unyielding basement walls. As a wall moves away, the soil expands laterally and shear strength is mobilised in its favour, dropping the pressure to the active value; as a wall pushes in, the soil is compressed and resists strongly, rising to the passive value.
Governing relations in practice
Rankine active and passive coefficients K_a = tan²(45 − φ/2) and K_p = tan²(45 + φ/2) are reciprocals for a level backfill, and passive pressure is far larger than active — for φ = 30°, K_a = 1/3 while K_p = 3, a nine-fold difference.
Design and analysis considerations
The thrust P_a = ½K_a γH² acts at H/3 above the base for a triangular pressure distribution with cohesionless level backfill; surcharge adds a rectangular pressure block and water adds a hydrostatic component that must be superposed.
Advanced theory and extensions
Cohesion reduces active pressure (2c√K_a term) producing tension near the top that soil cannot sustain, so a tension crack forms; the effective pressure diagram starts below the crack, and its neglect over-estimates the stabilising cohesion.
Assumptions and validity limits
State assumptions explicitly before using any relation for earth pressure theories — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Soil Mechanics viva and GATE descriptive questions, listing valid assumptions often earns separate marks.
Step-by-step problem approach
1. Read the question and list given data with SI units (common in Soil Mechanics papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to earth pressure theories.
4. Use equation 1:
5. Use equation 2:
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to earth pressure theories.
4. Use equation 1:
.
5. Use equation 2:
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.
Applications & exam relevance
Earth Pressure Theories appears in foundation and earthwork design. In Indian civil curricula this topic is tested because it connects theory to engineering properties of soils.
GATE and semester exams often combine earth pressure theories with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use earth pressure theories?" — answer with a lab, mini-project, or plant visit example if possible.
Common mistakes in exams
• Using the active coefficient where passive resistance is intended, or vice versa.
• Placing the resultant thrust at mid-height instead of H/3 for a triangular distribution.
• Ignoring the tension crack in cohesive backfill.
• Omitting the separate hydrostatic water pressure behind an undrained wall.
• Placing the resultant thrust at mid-height instead of H/3 for a triangular distribution.
• Ignoring the tension crack in cohesive backfill.
• Omitting the separate hydrostatic water pressure behind an undrained wall.
Quick revision checklist
Before attempting earth pressure theories problems, confirm you can:
1. Active: wall moves away; passive: wall moves toward soil
2. Cohesion: modify pressure diagram — tension crack possible at top
3. At-rest K_0 ≈ 1 − sin φ for normally consolidated soil
2. Cohesion: modify pressure diagram — tension crack possible at top
3. At-rest K_0 ≈ 1 − sin φ for normally consolidated soil
Revise the solved examples in Soil Mechanics & Foundations — BC Punmia and one previous-year GATE or university paper for this unit.
Worked examples
Try the problem first — open the solution when you are ready to check.
Active thrust on a retaining wall
Problem
A 5 m high retaining wall retains dry cohesionless backfill with φ = 30° and unit weight γ = 18 kN/m³. The backfill is level with no surcharge. Find the total active thrust and its point of application.
Solution
Active coefficient K_a = tan²(45 − 30/2) = tan²30° = 1/3. Total active thrust P_a = ½ K_a γ H² = ½ × (1/3) × 18 × 5² = ½ × (1/3) × 18 × 25 = 75 kN per metre run. It acts at H/3 = 5/3 = 1.67 m above the base, since the pressure distribution is triangular.
Conceptual check — Earth Pressure Theories
Problem
In a Soil Mechanics semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of earth pressure theories." What should a complete answer include?
Exams & GATE
BC Punmia — state Rankine vs Coulomb assumptions in answers.
📖 Standard books (India)
Soil Mechanics & Foundations — BC Punmia
Read: Syllabus unit
Soil properties, bearing capacity, and foundations
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